5 Reflectivity and amplitude variations with offset (AVO) in isotropic media
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97
3.5 Reflectivity and AVO in isotropic media
r1Ј V1
r2Ј V2
Figure 3.5.1 Reflection of a normal-incidence wave at an interface between two thick
homogeneous, isotropic, elastic layers.
IP ¼ VP
IS ¼ VS
where IP , IS are P- and S-wave impedances, VP , VS are P- and S-wave velocities, and
r is density.
At an interface between two thick homogeneous, isotropic, elastic layers, the
normal incidence reflectivity, defined as the ratio of the reflected wave amplitude
to the incident wave amplitude, is
RPP ¼
%
RSS ¼
%
2 VP2 À 1 VP1 IP2 IP1
ẳ
2 VP2 ỵ 1 VP1 IP2 ỵ IP1
1
lnIP2 =IP1 ị
2
2 VS2 1 VS1 IS2 IS1
ẳ
2 VS2 ỵ 1 VS1 IS2 ỵ IS1
1
lnIS2 =IS1 ị
2
where RPP is the normal incidence P-to-P reflectivity, RSS is the S-to-S reflectivity,
and the subscripts 1 and 2 refer to the first and second media, respectively
(Figure 3.5.1). The logarithmic approximation is reasonable for |R| < 0.5 (Castagna,
1993). A normally incident P-wave generates only reflected and transmitted P-waves.
A normally incident S-wave generates only reflected and transmitted S-waves. There
is no mode conversion.
AVO: amplitude variations with offset
For non-normal incidence, the situation is more complicated. An incident P-wave
generates reflected P- and S-waves and transmitted P- and S-waves. The reflection
98
Seismic wave propagation
VP1, VS1, r1
qS1
Reflected
S-wave
q1
Reflected
P-wave
Incident
P-wave
Transmitted
P-wave
q2
qS2
Transmitted
S-wave
VP2, VS2, r2
Figure 3.5.2 The angles of the incident, reflected, and transmitted rays of a P-wave
with non-normal incidence.
and transmission coefficients depend on the angle of incidence as well as on the
material properties of the two layers. An excellent review is given by Castagna
(1993).
The angles of the incident, reflected, and transmitted rays (Figure 3.5.2) are related
by Snell’s law as follows:
p¼
sin 1 sin 2 sin S1 sin S2
¼
¼
¼
VP1
VP2
VS1
VS2
where p is the ray parameter. y and yS are the angles of P- and S-wave propagation,
respectively, relative to the reflector normal. Subscripts 1 and 2 indicate angles or
material properties of layers 1 and 2, respectively.
The complete solution for the amplitudes of transmitted and reflected P- and
S-waves for both incident P- and S-waves is given by the Knott–Zoeppritz equations
(Knott, 1899; Zoeppritz, 1919; Aki and Richards, 1980; Castagna, 1993).
Aki and Richards (1980) give the results in the following convenient matrix form:
0
1
#"
""
#"
""
B PP SP PP SP C
B #" #" "" "" C
B PS SS PS SS C
B
C
B ## ## "# "# C ¼ MÀ1 N
B
C
B PP SP PP SP C
@
A
"#
"#
PS SS PS
##
##
SS
where each matrix element is a reflection or transmission coefficient for displacement
amplitudes. The first letter designates the type of incident wave, and the second letter
designates the type of reflected or transmitted wave. The arrows indicate downward #
and upward ↑ propagation, so that a combination #" indicates a reflection coefficient,
while a combination ## indicates a transmission coefficient. The matrices M and N
are given by
99
3.5 Reflectivity and AVO in isotropic media
0
À cos S1
sin 2
À sin S1
cos 2
À
Á
1 VS1 1 À 2 sin2 S1
22 VS2 sin S2 cos 2
1 VS1 sin 2S1
À
Á
2 VP2 1 À 2 sin2 S2
À sin 1
B
B
B
cos 1
B
M ¼B
B
B 21 VS1 sin S1 cos 1
B
@
À
Á
À1 VP1 1 À 2 sin2 S1
0
sin 1
B
B
B
cos 1
B
N ¼B
B
B 21 VS1 sin S1 cos 1
B
@
À
Á
1 VP1 1 À 2 sin2 S1
cos S2
1
C
C
C
C
C
À
Á
C
2 VS2 1 À 2 sin2 S2 C
C
A
À2 VS2 sin 2S2
À sin S2
cos S1
À sin 2
À cos S2
À sin S1
cos 2
À
Á
1 VS1 1 À 2 sin2 S1
22 VS2 sin S2 cos 2
À1 VS1 sin 2S1
À
Á
À2 VP2 1 À 2 sin2 S2
1
C
C
C
C
C
À
Á
C
2 VS2 1 À 2 sin2 S2 C
C
A
2 VS2 sin 2S2
À sin S2
Results for incident P and incident S, given explicitly by Aki and Richards (1980),
#"
#"
##
##
#"
#"
are as follows, where RPP ¼ PP; RPS ¼ PS; TPP ¼ PP; TPS ¼ PS; RSS ¼ SS; RSP ¼ SP;
##
##
TSP ¼ SP; TSS ¼ SS:
RPP
!.
cos 1
cos 2
cos 1 cos S2
2
ẳ b
c
F aỵd
Hp
D
VP1
VP2
VP1 VS2
RPS ẳ 2
!.
cos 1
cos 2 cos S2
ab ỵ cd
pVP1 VS1 Dị
VP1
VP2 VS2
TPP ẳ 21
cos 1
FVP1 =VP2 Dị
VP1
TPS ¼ 21
cos 1
HpVP1 =ðVS2 DÞ
VP1
RSS ¼ À
!.
cos S1
cos S2
cos 2 cos S1
E aỵd
Gp2 D
b
c
VS1
VS2
VP2 VS1
RSP
cos S1
cos 2 cos S2
pVS1 =VP1 Dị
ẳ2
ab ỵ cd
VS1
VP2 VS2
TSP ẳ 21
TSS ẳ 21
cos S1
GpVS1 =VP2 Dị
VS1
cos S1
EVS1 =VS2 Dị
VS1
100
Seismic wave propagation
where
a ẳ 2 1 À 2 sin2 S2 À 1 1 À 2 sin2 S1
b ẳ 2 1 2 sin2 S2 ỵ 21 sin2 S1
c ẳ 1 1 2 sin2 S1 ỵ 22 sin2 S2
À
Á
2
2
d ¼ 2 2 VS2
À 1 VS1
D ¼ EF ỵ GHp2 ẳ det Mị=VP1 VP2 VS1 VS2 ị
cos 1
cos 2
ỵc
VP1
VP2
cos S1
cos S2
Fẳb
ỵc
VS1
VS2
cos 1 cos S2
Gẳad
VP1 VS2
cos 2 cos S1
H ¼aÀd
VP2 VS1
E¼b
and p is the ray parameter.
Approximate forms
Although the complete Zoeppritz equations can be evaluated numerically, it is often
useful and more insightful to use one of the simpler approximations.
Bortfeld (1961) linearized the Zoeppritz equations by assuming small contrasts
between layer properties as follows:
!
1
VP2 2 cos 1
sin 1 2 2
lnð2 =1 ị
2
RPP 1 ị % ln
ỵ
VS1 VS2
ị 2ỵ
VP1 1 cos 2
VP1
2
lnðVS2 =VS1 Þ
Aki and Richards (1980) also derived a simplified form by assuming small layer
contrasts. The results are conveniently expressed in terms of contrasts in VP, VS, and
r as follows:
RPP ị %
1
1 VP
VS
ỵ
4p2 V"S2 "
1 4p2 V"S2
2
"
"
2
2 cos VP
VS
ÀpV"P
RPS ðÞ %
2 cos S
2 2
2 cos cos S
"
"
1 2VS p ỵ 2VS "
"
VP V"S
!
2 "2
2 cos cos S ÁVS
"
À 4p VS À 4VS "
VP V"S
V"S
1
1
VP
ỵ
1
TPP ị % 1
2
2 "
2 cos
V"P
101
3.5 Reflectivity and AVO in isotropic media
pV"P
TPS ðÞ %
2 cos S
2 2
2 cos cos S Á
"
"
1 À 2VS p À 2VS "
"
VP V"S
!
2 "2
2 cos cos S ÁVS
"
À 4p VS ỵ 4VS "
VP V"S
V"S
RSP ị%
cos S V"S
RPS ị
V"P cos
Á
1À
1
2 "2 Á
2 "2 ÁVS
À
À 4p VS
RSS ðÞ% À 1 À 4p VS
"
2
2 cos2 S
V"S
TSP ðÞ % À
cos S V"S
TPS
V"P cos
1
1
VS
ỵ
TSS ị % 1
1
2
2 "
2 cos S
V"S
where
pẳ
sin 1 sin S1
ẳ
VP1
VS1
ẳ 2 ỵ 1 ị=2
S ẳ S2 ỵ S1 ị=2
ẳ 2 1
" ẳ 2 ỵ 1 ị=2
VP ẳ VP2 VP1
V"P ẳ VP2 ỵ VP1 ị=2
VS ẳ VS2 VS1
V"S ẳ VS2 ỵ VS1 Þ=2
Often, the mean P-wave angle y is approximated as y1, the P-wave angle of
incidence.
The result for P-wave reflectivity can be rewritten in the familiar form:
RPP ị % RP0 ỵ B sin2 ỵ Ctan2 sin2 ị
or
!
1 VP
1 VP
VS
V"S2
ỵ
2 "2 2 " ỵ
ỵ
sin2
RPP ị %
2 V"P
"
2 V"P
"
VP
VS
ỵ
1 VP 2
2
tan
sin
2 V"P
This form can be interpreted in terms of different angular ranges (Castagna, 1993).
In the above equations RP0 is the normal incidence reflection coefficient as
expressed by
102
Seismic wave propagation
RP0 ẳ
IP2 IP1 IP 1 VP
%
%
ỵ
"
IP2 ỵ IP1
2IP
2 V"P
The parameter B describes the variation at intermediate offsets and is often called the
AVO gradient, and C dominates at far offsets near the critical angle.
Shuey (1985) presented a similar approximation where the AVO gradient is
expressed in terms of the Poisson ratio n as follows:
"
#
Á
Án
1 ÁVP À 2
2
RPP ð1 Þ % RP0 ỵ ERP0 ỵ
tan
sin
sin2 1 ỵ
1
1
2
2 V"P
1 "nị
where
1 VP
ỵ
RP0 %
"
2 V"P
1 2"n
E ẳ F 21 ỵ Fị
1 "n
Fẳ
VP =V"P
VP =V"P ỵ ="
and
n ẳ n2 n1
"n ẳ n2 ỵ n1 ị=2
The coefficients E and F used here in Shuey’s equation are not the same as those
defined earlier in the solutions to the Zoeppritz equations.
Smith and Gidlow (1987) offered a further simplification to the Aki–Richards
equation by removing the dependence on density using Gardner’s equation (see
Section 7.10) as follows:
/ V 1=4
giving
ÁVP
ÁVS
RPP ðÞ % c " þ d "
VP
VS
where
5 1 V"2
1
c ¼ À "S2 sin2 þ tan2
8 2 VP
2
V"2
d ¼ À 4 "S2 sin2
VP
103
3.5 Reflectivity and AVO in isotropic media
Wiggins et al. (1983) showed that when VP % 2VS, the AVO gradient is approximately (Spratt et al., 1993)
B % RP0 À 2RS0
given that the P and S normal incident reflection coefficients are
1 ÁVP
ỵ
RP0 %
"
2 V"P
1 VS
RS0 %
ỵ
"
2 V"S
Hilterman (1989) suggested the following slightly modified form:
RPP ị % RP0 cos2 ỵ PR sin2
where RP0 is the normal incidence reflection coefficient and
PR ẳ
n2 n1
1 "nị2
This modified form has the interpretation that the near-offset traces reveal the P-wave
impedance, and the intermediate-offset traces image contrasts in Poisson ratio
(Castagna, 1993).
Gray et al. (1999) derived linearized expressions for P–P reflectivity in terms of the
angle of incidence, y, and the contrast in bulk modulus, K, shear modulus, m, and bulk
density, r:
2
V"S
1 1 V"S2 2 K
1 2
2
Rị ẳ "2 sec
ỵ "2
sec 2 sin
4 3 VP
K
3
VP
1 1 2
ỵ
sec
2 4
Similarly, their expression in terms of Lames coefficient, l, shear modulus, and bulk
density is
2
V"S
1 1 V"S2 2 l
1 2
2
ỵ "2
sec 2 sin
Rị ẳ "2 sec
4 2 VP
l
2
VP
ỵ
1 1 2
Á
À sec
2 4
In the same assumptions of small layer contrast and limited angle of incidence, we
can write the linearized SV-to-SV reflection (Ruăger, 2001):
1 IS
7 VS
1 VS 2
RSV-iso S ị % " ỵ
ỵ2
sin S tan2 S
sin2 S
"
"
2 IS
2 VS
2 V"S
104
Seismic wave propagation
where yS is the SV-wave phase angle of incidence and IS ¼ VS is the shear
impedance.
Similarly, for SH-to-SH reflection (Ruăger, 2001):
RSH S ị ẳ
1 IS 1 VS
tan2 S
ỵ
2 I"S
2 V"S
where yS is the SH-wave phase angle of incidence.
For P-to-SV converted shear-wave reflection (Aki and Richards, 1980):
0
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1
2
À tan S @
Á
V"S
2
À sin2 S A
RPS ðS Þ % " " 1 2 sin S ỵ 2 cos S
"
"
2VS =VP
VP
0
s 1
2
V"S
tan S
VS
ỵ " " @4 sin2 S À 4 cos S
À sin2 S A "
V"P
VS
2VS =VP
where yS is the S-wave phase angle of reflection. This can be rewritten as (Gonza´lez,
2006)
RPS ðÞ %
À sin
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
ðV"P =V"S Þ ðV"P =V"S Þ À sin2
82
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3
2
< 1 V" 2
V"P
P
2
4
sin
ỵ
cos
sin2 5
"
"
: 2 VS
"
VS
9
s 3
2
=
"
VP
2 5 ÁVS
À42 sin2 À 2 cos
À
sin
V"S
V"S ;
2
where y is the P-wave phase angle of incidence.
For small angles, Duffaut et al. (2000) give the following expression for RPS(y):
!
V"S Á
V"S ÁVS
1
À 2 " " sin
RPS ị %
1ỵ2 "
2
VP "
VP VS
V"S
ỵ "
VP
!
1 V"S
V"S 1 2VS
ỵ "
sin3
"
ỵ
"
4 VP "
V"P 2
VS
which can be simplified further (Jı´lek, 2002b) to
!
1
V"S
V"S VS
RPS ị %
1ỵ2 "
2 " " sin
2
VP
VP VS
105
3.6 Plane-wave reflectivity in anisotropic media
Assumptions and limitations
The equations presented in this section apply in the following cases:
the rock is linear, isotropic, and elastic;
plane-wave propagation is assumed; and
most of the simplified forms assume small contrasts in material properties across the
boundary and angles of incidence of less than about 30 . The simplified form for
P-to-S reflection given by Gonza´lez (2006) is valid for large angles of incidence.
3.6
Plane-wave reflectivity in anisotropic media
Synopsis
An incident wave at a boundary between two anisotropic media (Figure 3.6.1) can
generate reflected quasi-P-waves and quasi-S-waves as well as transmitted quasiP-waves and quasi-S-waves (Auld, 1990). In general, the reflection and transmission
coefficients vary with offset and azimuth. The AVOA (amplitude variation with
offset and azimuth) can be detected by three-dimensional seismic surveys and is a
useful seismic attribute for reservoir characterization.
Brute-force modeling of AVOA by solving the Zoeppritz (1919) equations can be
complicated and unintuitive for several reasons: for anisotropic media in general, the
two shear waves are separate (shear-wave birefringence); the slowness surfaces are
nonspherical and are not necessarily convex; and the polarization vectors are neither
parallel nor perpendicular to the propagation vectors.
Schoenberg and Prota´zio (1992) give explicit solutions for the plane-wave reflection and transmission problem in terms of submatrices of the coefficient matrix of the
Zoeppritz equations. The most general case of the explicit solutions is applicable to
Reflected
qS-wave
Incident qP-wave
Reflected
qP-wave
q1
r1, a1, b1, e1, d1, g1
r2, a2, b2, e2, d2, g2
q2
Transmitted
qP-wave
Transmitted
qS-wave
Figure 3.6.1 Reflected and transmitted rays caused by a P-wave incident at a boundary between two
anisotropic media.
106
Seismic wave propagation
monoclinic media with a mirror plane of symmetry parallel to the reflecting plane.
Let R and T represent the reflection and transmission matrices, respectively,
2
RPP
6
R ¼4 RPS
RPT
2
TPP
T ¼ 4 TPS
TPT
RSP
RTP
3
RSS
7
RTS 5
RST
RTT
TSP
TSS
TST
3
TTP
TTS 5
TTT
where the first subscript denotes the type of incident wave and the second subscript
denotes the type of reflected or transmitted wave. For “weakly” anisotropic media, the
subscript P denotes the P-wave, S denotes one quasi-S-wave, and T denotes the other
quasi-S-wave (i.e., the tertiary or third wave). As a convention for real s23P , s23S , and s23T ,
s23P < s23S < s23T
where s3i is the vertical component of the phase slowness of the ith wave type when
the reflecting plane is horizontal. An imaginary value for any of the vertical slownesses implies that the corresponding wave is inhomogeneous or evanescent. The
impedance matrices are defined as
2
3
eP1
eS1
eT1
6
7
eP2
eS2
eT2
6
7
6
7
X ¼6 fÀðC13 eP1 þ C36 eP2 Þs1 fÀðC13 eS1 þ C36 eS2 Þs1 fC13 eT1 ỵ C36 eT2 ịs1 7
6
7
4 C23 eP2 ỵ C36 eP1 ịs2
C23 eS2 ỵ C36 eS1 ịs2
C23 eT2 ỵ C36 eT1 ịs2 5
C33 eP3 s3P g
2
fC55 s1 ỵ C45 s2 ịeP3
6 C55 eP1 ỵ C45 eP2 ịs3P g
6
6
6
Y ẳ 6 fC s ỵ C s ịe
45 1
44 2 P3
6
6 C e ỵ C e ịs g
45 P1
44 P2 3P
4
eP3
C33 eS3 s3S g
fC55 s1 ỵ C45 s2 ịeS3
C55 eS1 ỵ C45 eS2 ịs3S g
fC45 s1 ỵ C44 s2 ịeS3
C45 eS1 þ C44 eS2 Þs3S g
eS3
ÀC33 eT3 s3T g
3
fÀðC55 s1 þ C45 s2 ịeT3
C55 eT1 ỵ C45 eT2 ịs3T g 7
7
7
7
fC45 s1 ỵ C44 s2 ịeT3 7
7
C45 eT1 ỵ C44 eT2 Þs3T g 7
5
eT3
where s1 and s2 are the horizontal components of the phase slowness vector; eP, eS,
and eT are the associated eigenvectors evaluated from the Christoffel equations (see
Section 3.2), and CIJ denotes elements of the stiffness matrix of the incident medium.
X0 and Y0 are the same as above except that primed parameters (transmission
medium) replace unprimed parameters (incidence medium). When neither X nor Y
is singular and (X1X0 ỵ Y1Y0 ) is invertible, the reflection and transmission coefficients can be written as
107
3.6 Plane-wave reflectivity in anisotropic media
T ẳ 2X1 X0 ỵ Y1 Y0 ị1
R ẳ X1 X0 Y1 Y0 ịX1 X0 þ YÀ1 Y0 ÞÀ1
Schoenberg and Prota´zio (1992) point out that a singularity occurs at a horizontal
slowness for which an interface wave (e.g., a Stoneley wave) exists. When Y is
singular, straightforward matrix manipulations yield
T ẳ 2Y01 YX1 X0 Y01 Y ỵ Iị1
R ẳ X1 X0 Y01 Y ỵ Iị1 X1 X0 Y0À1 Y À IÞ
Similarly, T and R can also be written without X–1 when X is singular as
T ¼ 2X0À1 XI ỵ Y1 Y0 X01 Xị1
R ẳ I ỵ Y1 Y0 X0À1 XÞÀ1 ðI À YÀ1 Y0 X0À1 XÞ
Alternative solutions can be found by assuming that X0 and Y0 are invertible
R ẳ Y01 Y ỵ X01 Xị1 Y01 Y X01 Xị
T ẳ 2X01 XY01 Y ỵ X01 Xị1 Y01 Y
ẳ 2Y01 YY01 Y ỵ X01 Xị1 X01 X
These formulas allow more straightforward calculations when the media have at
least monoclinic symmetry with a horizontal symmetry plane.
For a wave traveling in anisotropic media, there will generally be out-of-plane
motion unless the wave path is in a symmetry plane. These symmetry planes include
all vertical planes in VTI (transversely isotropic with vertical symmetry axis) media
and the symmetry planes in HTI (transversely isotropic with horizontal symmetry
axis) and orthorhombic media. In this case, the quasi-P- and the quasi-S-waves in the
symmetry plane uncouple from the quasi-S-wave polarized transversely to the
symmetry plane. For weakly anisotropic media, we can use simple analytical formulas
(Banik, 1987; Thomsen, 1993; Chen, 1995; Ruăger, 1995, 1996) to compute AVOA
(amplitude variation with offset and azimuth) responses at the interface of anisotropic
media that can be either VTI, HTI, or orthorhombic. The analytical formulas give
more insight into the dependence of AVOA on anisotropy. Vavrycˇuk and Psˇencˇ´ık
(1998) and Psˇencˇ´ık and Martins (2001) provide formulas for arbitrary weak
anisotropy.